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Non-weak compactness of the integration map for vector measures

Part of: Set functions, measures and integrals with values in abstract spaces Integral operators

Published online by Cambridge University Press:  09 April 2009

S. Okada  and
W. J. Ricker
S. Okada
Affiliation:
University of Tasmania, Hobart, Tas., 7001, Australia
W. J. Ricker
Affiliation:
University of New South Wales, Kensington, NSW, 2033, Australia
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Abstract

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Let m be a vector measure with values in a Banach space X. If L1(m) denotes the space of all m integrable functions then, with respect to the mean convergence topology, L1(m) is a Banach space. A natural operator associated with m is its integration map Im which sends each f of L1(m) to the element ∫fdm (of X). Many properties of the (continuous) operator Im are closely related to the nature of the space L1(m). In general, it is difficult to identify L1(m). We aim to exhibit non-trivial examples of measures m in (non-reflexive) spaces X for which L1(m) can be explicitly computed and such that Im is not weakly compact. The examples include some well known operators from analysis (the Fourier transform on L1 ([−π, π]), the Volterra operator on L1 ([0, 1]), compact self-adjoint operators in a Hilbert space); such operators can be identified with integration maps Im (or their restrictions) for suitable measures m.

MSC classification

Secondary: 28B05: Vector-valued set functions, measures and integrals 45P05: Integral operators
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 54 , Issue 3 , June 1993 , pp. 287 - 303
Copyright
Copyright © Australian Mathematical Society 1993

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